Graph the function \( f(x) = \sqrt[3]{x - 4} \) and identify its domain and range.

Certainly! Let's analyze and graph the function \( f(x) = \sqrt[3]{x - 4} \), and identify its domain and range. ### Function Analysis 1. **Basic Cube Root Function**: - The standard cube root function is \( f(x) = \sqrt[3]{x} \). - Its graph passes through the origin (0,0), with symmetry about the origin. - It increases continuously from left to right, extending to both positive and negative infinity. 2. **Transformation Applied**: - The function \( f(x) = \sqrt[3]{x - 4} \) involves a horizontal shift. - Subtracting 4 inside the cube root shifts the graph **4 units to the right**. - So, instead of passing through (0,0), it passes through (4,0). ### Graph of \( f(x) = \sqrt[3]{x - 4} \) Below is a sketch of the graph: ``` y | | / | / | / | / |------------(4,0)----------- | / | / | / | / | / | / | +---------------------------- x ``` **Key Features**: - **Point of Shift**: The graph passes through the point (4,0). - **Behavior**: - As \( x \) approaches infinity, \( f(x) \) also approaches infinity. - As \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity. - The graph is smooth and has no breaks or sharp corners. ### Domain and Range - **Domain**: - The domain of a cube root function is all real numbers because you can take the cube root of any real number (positive, negative, or zero). - **Domain**: \( (-\infty, \infty) \) - **Range**: - Similarly, the range is all real numbers since the output of the cube root function can be any real number. - **Range**: \( (-\infty, \infty) \) ### Summary - **Graph**: A shifted cube root curve passing through (4,0), increasing from left to right without any restrictions. - **Domain**: All real numbers \( \mathbb{R} \) or \( (-\infty, \infty) \). - **Range**: All real numbers \( \mathbb{R} \) or \( (-\infty, \infty) \). If you have access to graphing tools like Desmos, GeoGebra, or a graphing calculator, you can plot the function \( f(x) = \sqrt[3]{x - 4} \) to visualize its shape more accurately.